Inexact Bregman iteration for deconvolution of superimposed extended and point sources

• The deconvolution of high contrast images consisting of very bright stars and smooth structures around the stars is studied.
• To restore the region around the stars, the object can be regarded as superposition of point source and extended source.
• When the position of the point source is known, we introduce a regularization term only for the diffuse component.
• We give conditions for the solution of the related variational problem for Poisson data with Tikhonov-like regularization.
• In presence of an overestimation of the regularization parameter, we solve by the inexact Bregman iteration with SGP method.

In this paper we consider the deconvolution of high contrast images consisting of very bright stars (point component) and smooth structures underlying the stars (diffuse component). A typical case is a weak diffuse jet line emission superimposed to a strong stellar continuum. In order to reconstruct the diffuse component, the original object can be regarded as the sum of these two components. When the position of the point sources is known, a regularization term can be introduced for the second component. An approximation of the original object can be obtained by solving a reduced variational problem whose unknowns are the intensities of the stars and the diffuse component. We analyze this problem when the detected image is corrupted by Poisson noise and Tikhonov-like regularization is used, giving conditions for the existence and the uniqueness of the solution. Furthermore, since only an overestimation of the regularization parameter is available, we propose to solve the variational problem by inexact Bregman iteration combined with a Scaled Gradient Projection method (SGP). Numerical simulations show that the images obtained with this approach enable us to reconstruct the original intensity distribution around the point source with satisfactory accuracy.